\(P=\frac{25}{x+5}-\frac{1}{x-2}=\frac{25}{x+5}-\frac{-1}{-\left(x-2\right)}=\frac{25}{x+5}+\frac{1}{2-x}\)
Áp dụng bđt Cauchy-Schwarz dạng Engel ta có:
\(P=\frac{5^2}{x+5}+\frac{1^2}{2-x}\ge\frac{\left(5+1\right)^2}{x+5+2-x}=\frac{6^2}{7}=\frac{36}{7}\)
Dấu "=" xảy ra khi \(\frac{5}{x+5}=\frac{1}{2-x}\)\(\Leftrightarrow5\left(2-x\right)=x+5\)
\(\Leftrightarrow10-5x=x+5\Leftrightarrow5=6x\Leftrightarrow x=\frac{5}{6}\left(TM\right)\)
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